For a transmission line if \(\frac{L}{C} = \frac{R}{G}\) then which of the following is incorrect?

Concept:

The characteristic impedance of a transmission line is defined as:

\({Z_0} = \sqrt {\frac{{R' + j\omega L'}}{{G' + j\omega C'}}} \)    ---(1)

And the propagation constant of the transmission line is defined as:

\(\gamma = \alpha + j\beta = \sqrt {\left( {R' + j\omega C'} \right)\left( {G' + j\omega C'} \right)} \)    ---(2)

Where,

α is attenuation constant

β is phase constant

R’ = Resistance per unit length of the line

G’ is conductance per unit length of the line

L’ is the inductance per unit length of the line

C’ is the capacitance per unit length of the line

Analysis:

A distortion less line satisfies the following condition:

\(\frac{{R'}}{{L'}} = \frac{{G'}}{{C'}}\)

So, the characteristic impedance of the distortionless line will be:

\({Z_0} = \sqrt {\frac{{L'}}{{C'}}} = \sqrt {\frac{{R'}}{{G'}}} \)

∴ The characteristic impedance of both lossless and a distortionless line is real.

And the propagation constant of the distortion less line will be:

\(\gamma = \alpha + j\beta = \sqrt {R'G'} + j\omega \sqrt {L'C'} \)

\(\alpha = \sqrt {R'G'} \ne 0\)

Therefore, the attenuation constant of distortion less line is not zero but it is real.

As the line is distortionless, if a series of pulses are transmitted they arrive undistorted.

Important Points

For a lossless line:

R’ = G’ = 0

So, the characteristic impedance of a lossless transmission line using Equation (1) will be:

\({Z_0} = \sqrt {\frac{{L'}}{{C'}}} \)

And the propagation constant of a lossless transmission line using Equation (2) will be:

\(\gamma = \alpha + j\beta = j\omega \sqrt {L'C'} \)

α = 0

Therefore, the attenuation constant of the lossless line is always zero (real).